This rule has been implicitly tested countless times since it enters literally every calculation in which quantum effects are relevant. But it is not usually tested for parameterized deviations like, say, Einstein's field equations are tested for such deviations. Now however, a group of physicists (from the Institute for Quantum Computing and Perimeter Institute in Waterloo, Canada, the Laboratoire the Nanotechlogie et d'Instrumentation Optique in Troye, France, and the Institut für Experimentalphysik in Innsbruck, Austria) has tested Born's rule for deviations stemming from higher order interference which serves to constrain possible modifications of quantum mechanics. Their results were published in a recent Science issue:
- Ruling Out Multi-Order Interference in Quantum Mechanics
Urbasi Sinha, Christophe Couteau, Thomas Jennewein, Raymond Laflamme, Gregor Weihs
Science 23 July 2010: Vol. 329. no. 5990, pp. 418 - 421
The short summary is that they haven't found any deviation to a precision of one in a hundred. But their method is really neat and worth spending a paragraph on.
The experimental setup that the group has used is a tripe-slit through which pass single photons. If one computes the probability to measure a photon at a particular location on the detector screen in usual quantum mechanics, you square the sum of the wave-functions originating from each of the three slits. You get several mixed terms, but they are all second order in the wave-function. If Born's rule holds, this allows you to express the probability for the three-slit experiment as a sum of probabilities from leaving open only one of the slits and leaving open combinations of two slits. Thus, what the clever experimentalist do is a series of measurements leaving each single slit open, all combinations of two slits open, and leaving all three slits open, and see if the probabilities add up. And they do, to very good precision.
So, there's nothing groundbreaking to report here in terms of novel discoveries, but I very much like the direct test of the foundations of quantum mechanics this experiment constitutes. I think we could use more tests in this direction, and higher precision will come with time.